Sampling sequences for \(A^{-\infty}\) (Q1371334)

For every real \(n>0\), the space \(A^{-n}\) consists of all functions analytic in the unit disc \(\mathbb{D}\) and satisfying \(\sup_{z\in\mathbb{D}}|f(z)|(1-|z|^2)^n< \infty\). A set \(E\subset\mathbb{D}\) is said to be sampling for \(A^{-n}\) if the latter supremum restricted to \(z\in E\) defines an equivalent norm on \(A^{-n}\). Let \(A^{-\infty}= \bigcup_{n> 0}A^{-n}\). For \(f\in A^{-\infty}\) and \(E\subset\mathbb{D}\), we put \(T(f)= \inf\{n:f\in A^{-n}\}\) and \[ T_E(f)= \inf\Biggl\{n: \sup_{z\in E}|f(z)|(1-|z|^2)^n< \infty\Biggl\}. \] A set \(E\subset\mathbb{D}\) is said to be sampling for \(A^{-\infty}\) if \(T_E(f)= T(f)\). One of the main results of the paper says that if a sequence \(E\) is sampling for all \(A^{-n}\), then so it is for \(A^{-\infty}\), but the converse fails in general. Some specific necessary and (in general, different) sufficient conditions for a set to be sampling for \(A^{-\infty}\) are discussed.